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\section{Conclusions}  We have presented a new approach for calculating matrices. This method is suitable for applications where the cost of computing each matrix element is high in comparison to the cost of linear algebra operations. Our approach leverages the power of compressed sensing to avoid computing every matrix element individually, element,  thereby achieving substantial computational savings. When applied to the problem of molecular vibrations of organic molecules, our method results in accurate frequencies and normal modes with about 30\% of the expensive quantum mechanical computations usually required, which represents a quite significant 3x speed-up. Depending on the sparsity of the Hessian, we have shown that our method can improve the overall scaling of the computation.  Our It is interesting to be applied to other calculations where calculations that are common in the computational chemistry, for example the Fock matrix or the reponse matrix in linear-response time-dependent DFT. Nevertheless, our  method is general, however, not restricted to quantum chemistry  and it is applicable to many problems throughout the physical sciences and beyond. The main requirement is an \emph{a priori} guess of a basis in which the matrix to be computed is sparse. The optimal way to achieve this requirement is problem-dependent, but as research into sparsifying transformations continues to develop, we believe our method will enable considerable computational savings in a diverse array of scientific fields. In fact, a recent area of interest in compressed sensing is to develop dictionary learning methods that do not directly require the knowledge of a sparsifying basis, but that generate it on-the-fly based on the problem~\cite{Aharon_2006,Rubinstein_2010}. We believe that combining our matrix recovery protocol with state-of-the-art dictionary learning methods may eventually result in a new computational paradigm for the calculation of scientific matrices.